Non-Local Gravity: Theory & Applications

• Non-Local Gravity is a family of theories where gravitational interactions depend on a causal spacetime history, introducing effective dark matter without new particles.
• It is formulated within a teleparallel framework that replaces curvature with torsion, modifying the constitutive relations through nonlocal integral kernels.
• The theory offers testable predictions across galactic rotation curves, cosmological evolution, and solar system dynamics, challenging standard dark matter paradigms.

Non-Local Gravity (NLG) denotes a family of gravitational theories in which the gravitational response is not strictly pointwise but depends on a causal spacetime history or on non-local operators. In the best-developed usage surveyed here, NLG is a classical nonlocal generalization of Einstein gravity built in a teleparallel framework and modeled on nonlocal electrodynamics of media; its defining claim is that nonlocality can simulate dark matter through an effective source term rather than through new particles (Mashhoon, 2011). In other parts of the literature, the same label also covers theories with inverse d’Alembertian operators such as $f(R,\Box^{-1}R)$ and theories with analytic form factors built from $\Box$, where the emphasis shifts toward cosmology, ultraviolet behavior, localization methods, or compact-object solutions (Capozziello et al., 2022, Calcagni et al., 2018).

1. Conceptual scope and geometric formulations

The teleparallel NLG program starts from the teleparallel equivalent of general relativity (TEGR), where the gravitational variables are tetrads and the Weitzenböck connection, so that gravity is encoded by torsion rather than curvature. In that setting, the local constitutive relation between torsion and gravitational excitation is replaced by a causal integral relation, in close analogy with nonlocal electrodynamics of media (Mashhoon, 2011, Mashhoon, 2014). The preferred tetrad field is central in this construction: the theory has $16$ tetrad components, $10$ determining the metric and $6$ corresponding to local Lorentz gauge freedom, and the reformulated constitutive law is designed so that the nonlocality acts directly on quantities measured by preferred observers (Mashhoon, 2022).

In the linear approximation around Minkowski spacetime, this teleparallel construction can be rewritten as linearized general relativity sourced by ordinary matter plus an effective dark component. The key structural relation is

$$G_{ij}(x) + \int K(x,y)\,G_{ij}(y)\,d^4y = \kappa T_{ij}(x),$$

which, after introducing the reciprocal kernel, becomes

$$G_{ij} = \kappa \bigl(T_{ij} + T^D_{ij}\bigr), \qquad T^D_{ij}(x) = \int R(x,y)\,T_{ij}(y)\,d^4y.$$

In this form, nonlocality is mathematically equivalent to an additional source term and is interpreted as effective dark matter rather than as an extra matter species (0812.1059).

The same corpus also contains a local limit of teleparallel NLG. There the nonlocal kernel is collapsed to a local susceptibility $S(x)$, with general relativity recovered for $S=0$ and the constitutive law modified by spacetime-dependent response coefficients (Tabatabaei et al., 2022, Tabatabaei et al., 2023). Separately, other non-local gravity programs use inverse d’Alembertian operators or analytic form factors instead of a constitutive kernel; these are usually motivated by cosmology, effective actions, renormalizability, or the Cauchy problem rather than by dark-matter phenomenology alone (Capozziello et al., 2022, Calcagni et al., 2018).

Usage of “NLG”	Nonlocal ingredient	Representative consequence
Teleparallel constitutive NLG	Causal kernel in the constitutive relation	Effective dark matter in the Newtonian limit
Local limit of teleparallel NLG	Susceptibility $S(x)$	Modified flat cosmology; de Sitter excluded for dynamic $\Box$0
Inverse-operator or form-factor models	$\Box$1 or analytic functions of $\Box$2	Exact cosmological solutions, localization methods, or wormholes

A recurring misconception is to treat all of these constructions as interchangeable. The literature does not support that identification. The teleparallel constitutive theory, the local-limit susceptibility theory, inverse-$\Box$3 cosmologies, and analytic-form-factor models share the word “nonlocal,” but they are formulated with different dynamical variables, different constitutive or action principles, and different phenomenological targets (Mashhoon, 2022, Capozziello et al., 2022, Calcagni et al., 2018).

2. Newtonian regime and the effective dark-matter mechanism

The Newtonian limit is the most explicit and phenomenologically mature sector of teleparallel NLG. The gravitational force on a test mass remains

$\Box$4

but the potential obeys a nonlocal Poisson equation,

$\Box$5

which can be rewritten in reciprocal form as

$\Box$6

In Fourier space,

$\Box$7

so the nonlocal term acts exactly like an additional mass distribution (Roshan et al., 2021). A fundamental point emphasized in this formulation is that $\Box$8 if $\Box$9; there is no dark component without ordinary matter to generate it (Roshan et al., 2021).

The observationally motivated reciprocal kernels are

$16$0

with fitted galactic-scale parameters

$16$1

The inverse scale $16$2 kpc marks the distance beyond which the effective coupling saturates (Roshan et al., 2021). In the weak-field acceleration law used in galaxy applications, the kernel is often written in the simpler $16$3 form

$16$4

which leads to

$16$5

for NLG (Haghighi et al., 2016).

For a point mass $16$6, the nonlocal Poisson equation becomes

$16$7

so each baryonic point source is accompanied by a spherical cloud of effective dark matter of density $16$8. The total effective dark mass is

$16$9

with $10$0 in the simplest case, so the effective dark mass is roughly $10$1 (Roshan et al., 2021). In finite systems, however, only part of the halo is enclosed. If the baryonic diameter is $10$2,

$10$3

This yields the scale dependence emphasized in the galactic literature: globular clusters should have little effective dark matter, dwarf galaxies may have only modest amounts, and giant spirals or clusters can show strong dark-matter-like behavior (Roshan et al., 2021).

The effective dark matter generated by this convolution is also qualitatively constrained. It is described as smooth, positive, spherically symmetric, and independent of the matter density itself once the kernel is fixed; as a consequence, the effective density is finite and cusp-free (Roshan et al., 2022). That prediction is nontrivial observationally: it aligns with claims of low effective dark matter in some ultra-diffuse galaxies, but it struggles with systems whose inner rotation curves are too steep for a smooth central profile (Roshan et al., 2022).

3. Galaxy dynamics, rotation curves, and dwarf-galaxy phenomenology

Galaxy-scale tests dominate the phenomenology of teleparallel NLG. In the LITTLE THINGS analysis of $10$4 dwarf galaxies, the universal NLG parameters were kept fixed at

$10$5

and only the stellar mass-to-light ratio $10$6 was varied. The discretized radial acceleration was computed under cylindrical symmetry and inserted into the rotation-curve model, with the fit based on

$10$7

The reported average goodness-of-fit values were

$10$8

while the average stellar mass-to-light ratios were

$10$9

In that sample, NLG fit the dwarf-galaxy rotation curves reasonably well, but did so with stellar mass-to-light ratios larger than standard Milky Way values (Haghighi et al., 2016).

A second line of evidence comes from $6$0-body simulations of spiral galaxies. In the comparison between an exponential disk in NLG (ENLG) and a Newtonian exponential disk embedded in a live Plummer halo (EPL), both systems were initialized with the same baryonic distribution, particle velocities, random velocities, and Toomre parameter. With $6$1 and $6$2, the initial rotation curves were made nearly identical, but the subsequent evolution differed substantially. The bar instability growth rates were reported as

$6$3

so the NLG bar grew about $6$4 faster than the standard dark-matter case and about $6$5 faster than MOG. At late times the NLG bars were weaker and faster, and the ratio

$6$6

approached the fast-bar boundary $6$7, unlike earlier EPL simulations that reached $6$8 (Roshan et al., 2019).

The ultra-diffuse galaxy analysis sharpened the cusp-free prediction. AGC 114905 was found to be broadly consistent with the low-effective-dark-matter expectation of NLG, whereas AGC 242019 and AGC 219533 were identified as challenging cases because NLG could not explain the sharp slopes of the rotation curves at small radii (Roshan et al., 2022). This contrast is significant because it turns the smooth-kernel construction into an observational discriminator rather than a merely formal property.

The most recent dwarf-spheroidal study extended the testing program from rotation curves to Jeans modeling of line-of-sight velocity dispersions in Carina, Draco, Fornax, Leo I, Leo II, Sculptor, Sextans, and Ursa Minor. The model used the Plummer profile for the baryonic stellar density, a constant anisotropy parameter $6$9, and an emcee MCMC fit over

$$G_{ij}(x) + \int K(x,y)\,G_{ij}(y)\,d^4y = \kappa T_{ij}(x),$$0

The joint analysis of all eight galaxies yielded

$$G_{ij}(x) + \int K(x,y)\,G_{ij}(y)\,d^4y = \kappa T_{ij}(x),$$1

while Fornax and Sextans showed the main parameter inconsistency, with $$G_{ij}(x) + \int K(x,y)\,G_{ij}(y)\,d^4y = \kappa T_{ij}(x),$$2 only marginally compatible with previous literature at about the $$G_{ij}(x) + \int K(x,y)\,G_{ij}(y)\,d^4y = \kappa T_{ij}(x),$$3 level (Martino et al., 29 Jul 2025). This suggests that NLG remains phenomenologically viable in dwarf-galaxy kinematics, but not parameter-free in practice.

4. Dynamical friction, gravitational scattering, and barred spirals

A major technical result of teleparallel NLG is the derivation of a Chandrasekhar-type dynamical friction law in the Newtonian regime. In the approximation used for gravitational encounters, the nonlocal force enhancement is represented by

$$G_{ij}(x) + \int K(x,y)\,G_{ij}(y)\,d^4y = \kappa T_{ij}(x),$$4

so that the change in the velocity of the massive body $$G_{ij}(x) + \int K(x,y)\,G_{ij}(y)\,d^4y = \kappa T_{ij}(x),$$5 in a single encounter is computed with

$$G_{ij}(x) + \int K(x,y)\,G_{ij}(y)\,d^4y = \kappa T_{ij}(x),$$6

After integrating over a background distribution, the NLG analogue of Chandrasekhar’s formula becomes

$$G_{ij}(x) + \int K(x,y)\,G_{ij}(y)\,d^4y = \kappa T_{ij}(x),$$7

and for $$G_{ij}(x) + \int K(x,y)\,G_{ij}(y)\,d^4y = \kappa T_{ij}(x),$$8,

$$G_{ij}(x) + \int K(x,y)\,G_{ij}(y)\,d^4y = \kappa T_{ij}(x),$$9

A parallel derivation using the gravitational wake yields the same structure, with the nonlocality entering through a squared enhancement of the gravitational scattering strength (Roshan et al., 2021).

The enhancement is strongly system dependent. The estimates reported were

$$G_{ij} = \kappa \bigl(T_{ij} + T^D_{ij}\bigr), \qquad T^D_{ij}(x) = \int R(x,y)\,T_{ij}(y)\,d^4y.$$0

with similar values for the Fourier-space analogue $$G_{ij} = \kappa \bigl(T_{ij} + T^D_{ij}\bigr), \qquad T^D_{ij}(x) = \int R(x,y)\,T_{ij}(y)\,d^4y.$$1 (Roshan et al., 2021). This scale dependence is consistent with the earlier halo-enclosure argument: compact systems receive negligible nonlocal friction, while galaxy-scale systems exhibit stronger effective coupling.

The barred-galaxy consequence is especially important. In a particle dark-matter halo, the bar transfers angular momentum to the halo and strong dynamical friction leads to slow bars. In NLG there is no dark halo of particles; the bar interacts only with baryonic matter through the enhanced but still modest NLG friction. Using a disk thickness $$G_{ij} = \kappa \bigl(T_{ij} + T^D_{ij}\bigr), \qquad T^D_{ij}(x) = \int R(x,y)\,T_{ij}(y)\,d^4y.$$2 kpc, the estimated enhancement was

$$G_{ij} = \kappa \bigl(T_{ij} + T^D_{ij}\bigr), \qquad T^D_{ij}(x) = \int R(x,y)\,T_{ij}(y)\,d^4y.$$3

That increase is far too small to reproduce the strong braking effect of a particle dark halo, and NLG therefore predicts weak bar slowdown and the persistence of fast barred spirals (Roshan et al., 2021). This is not a secondary detail but a structural difference between effective nonlocal dark matter and a live particulate halo.

5. Solar-system regime, gravitomagnetism, and other weak-field tests

The solar-system program asks whether the same reciprocal kernels used on galactic scales are compatible with precision weak-field measurements. The reciprocal kernel family introduces a short-range regularization scale $$G_{ij} = \kappa \bigl(T_{ij} + T^D_{ij}\bigr), \qquad T^D_{ij}(x) = \int R(x,y)\,T_{ij}(y)\,d^4y.$$4 in addition to $$G_{ij} = \kappa \bigl(T_{ij} + T^D_{ij}\bigr), \qquad T^D_{ij}(x) = \int R(x,y)\,T_{ij}(y)\,d^4y.$$5 and $$G_{ij} = \kappa \bigl(T_{ij} + T^D_{ij}\bigr), \qquad T^D_{ij}(x) = \int R(x,y)\,T_{ij}(y)\,d^4y.$$6: $$G_{ij} = \kappa \bigl(T_{ij} + T^D_{ij}\bigr), \qquad T^D_{ij}(x) = \int R(x,y)\,T_{ij}(y)\,d^4y.$$7 with

$$G_{ij} = \kappa \bigl(T_{ij} + T^D_{ij}\bigr), \qquad T^D_{ij}(x) = \int R(x,y)\,T_{ij}(y)\,d^4y.$$8

recovered when $$G_{ij} = \kappa \bigl(T_{ij} + T^D_{ij}\bigr), \qquad T^D_{ij}(x) = \int R(x,y)\,T_{ij}(y)\,d^4y.$$9 (Roshan et al., 2022). The dimensionless parameter

$S(x)$0

controls the asymptotic enhancement (Roshan et al., 2022, Chicone et al., 2015).

In the solar system, the relevant regime satisfies $S(x)$1 kpc, so the nonlocal correction to the inverse-square law can be expanded in powers of $S(x)$2. The dominant magnitude for a spherical source at distance $S(x)$3 is controlled by

$S(x)$4

At $S(x)$5 AU, if $S(x)$6 AU and $S(x)$7 kpc, then

$S(x)$8

which places the predicted deviation in the range of proposed inverse-square-law experiments (Roshan et al., 2022).

Perihelion precession provides the strongest current bound. Updated Saturn ephemerides imply

$S(x)$9

improving the earlier bound by about a factor of $S=0$0 (Roshan et al., 2022). The preliminary earlier estimates from planetary precession were

$S=0$1

from Saturn, with weaker Mercury bounds (Chicone et al., 2015).

Other standard weak-field observables are much less constraining. For light rays grazing the Sun, the nonlocal correction to the deflection angle was found to be of order $S=0$2, and the corresponding gravitational time-delay correction was likewise negligible (Chicone et al., 2015). In the linearized theory more generally, NLG preserves causality through Volterra kernels and admits a weak-field lensing formalism for uniformly moving sources, with nonlocal corrections entering the deflection law through the Yukawa-type modification (Mashhoon, 2014).

The stationary, weak-field, slow-motion limit also yields a nonlocal gravitoelectromagnetic sector. For an isolated rotating source, the gravitomagnetic vector potential and field retain the usual dipole-like structure of general relativity plus a nonlocal correction proportional to a galactic-scale length, and the correction around Earth was estimated to be at least ten orders of magnitude smaller than the GR prediction (Mashhoon et al., 2019). NLG therefore reproduces the qualitative weak-field gravitomagnetic phenomenology of GR while shifting appreciable departures to galactic scales.

6. Cosmology, local limits, and broader non-local gravity programs

The full teleparallel NLG field equations are highly nonlinear, and exact cosmological solutions of the nonlocal theory are not known. This motivates the study of the local limit, in which the kernel is replaced by a susceptibility function $S=0$3 and the constitutive law becomes local but spacetime dependent (Tabatabaei et al., 2023, Tabatabaei et al., 2023). In homogeneous and isotropic cosmology with $S=0$4, the modified flat Friedmann equations are

$S=0$5

$S=0$6

with continuity relation

$S=0$7

Only spatially flat FLRW cosmologies are allowed in this local-limit model, and de Sitter spacetime is not a solution unless $S=0$8 (Tabatabaei et al., 2023).

The more detailed teleparallel constitutive analysis reaches the same conclusion in a broader exact-FLRW setting. For $S=0$9, the off-diagonal field equations force $S(x)$0 to be constant, whereas a genuinely time-dependent susceptibility is compatible only with the modified Cartesian flat model (Tabatabaei et al., 2022). In that model the densities evolve as

$S(x)$1

so even a $S(x)$2 component becomes dynamic unless $S(x)$3 is constant (Tabatabaei et al., 2022). Using the parametrization $S(x)$4, the model was reported to yield $S(x)$5 for CMB+BAO and $S(x)$6 for CMB+BAO+R19, thereby alleviating the $S(x)$7 tension (Tabatabaei et al., 2022).

The anisotropic local-limit program extends these results to Bianchi type I cosmologies. With

$S(x)$8

the susceptibility modifies both the evolution equations and the continuity equation, and the novel behavior arises specifically from $S(x)$9 (Tabatabaei et al., 2023). In this setting, de Sitter and Kasner spacetimes both fail unless $\Box$00 is constant, while the directional deceleration parameters acquire explicit $\Box$01-dependent terms and can exhibit anisotropic acceleration (Tabatabaei et al., 2023). The local limit therefore functions less as a dark-matter substitute than as an effective constitutive cosmology with dynamic dark-energy-like behavior.

Beyond teleparallel NLG, the same terminology is used for several other non-local gravity programs. In analytic-form-factor gravity, the action contains terms such as

$\Box$02

and the diffusion-equation method converts the dynamics into a localized $\Box$03-dimensional system. In that framework, the original non-local gravity theory requires $\Box$04 initial conditions and has $\Box$05 non-perturbative degrees of freedom in $\Box$06 (Calcagni et al., 2018). In the Palatini formulation with analytic form factors, a different conclusion emerges: for the special case $\Box$07, vacuum solutions of GR remain solutions of the non-local model, so singularities of Einstein gravity are not removed in that sector (Briscese et al., 2015).

Inverse-$\Box$08 cosmological models form another distinct branch. The survey literature discusses $\Box$09, $\Box$10, and $\Box$11 theories, selected in part through Noether symmetries, with exact de Sitter and power-law solutions and astrophysical constraints from the S2 star orbit around SgrA$\Box$12. In the cited spherical analysis, the physical non-locality scale $\Box$13 was constrained to roughly $\Box$14-$\Box$15 AU, with an example best fit near $\Box$16 AU (Capozziello et al., 2022). In non-local integral-kernel theories of the form

$\Box$17

the non-local geometric sector can support traversable wormholes without exotic matter; in the most favorable case, the linear model $\Box$18 with a power-law scalar ansatz yields stable and traversable wormholes satisfying asymptotic flatness and flaring-out conditions (Capozziello et al., 2022).

Taken together, these results show that “Non-Local Gravity” is not a single model but a research domain organized around different implementations of nonlocality. The teleparallel constitutive theory is the branch most directly associated with effective dark matter and galactic phenomenology; the local-limit susceptibility theory reshapes late-time cosmology; analytic-form-factor theories emphasize localization and degrees of freedom; Palatini models expose limitations of singularity resolution; and inverse-$\Box$19 models support distinct cosmological and compact-object constructions (Roshan et al., 2021, Tabatabaei et al., 2022, Calcagni et al., 2018, Briscese et al., 2015, Capozziello et al., 2022).